Solve The Inequality: ${\frac{1}{3}b - 4 \leq 4}$Enter The Correct Answer.

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Introduction

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In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems in algebra, geometry, and other branches of mathematics. Inequalities are used to compare two or more quantities, and they can be either linear or non-linear. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of an equation with a variable and a constant on one side of the inequality sign.

What is a Linear Inequality?

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A linear inequality is an inequality that can be written in the form of an equation with a variable and a constant on one side of the inequality sign. For example, the inequality $\frac{1}{3}b - 4 \leq 4$ is a linear inequality because it can be written in the form of an equation with a variable (b) and a constant (4) on one side of the inequality sign.

Solving Linear Inequalities

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To solve a linear inequality, we need to isolate the variable on one side of the inequality sign. This can be done by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value. However, we need to be careful when multiplying or dividing both sides of the inequality by a negative value, as this can change the direction of the inequality.

Adding and Subtracting

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When adding or subtracting both sides of the inequality by the same value, the direction of the inequality remains the same. For example, if we have the inequality $\frac{1}{3}b - 4 \leq 4$, we can add 4 to both sides of the inequality to get $\frac{1}{3}b \leq 8$.

Multiplying and Dividing

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When multiplying or dividing both sides of the inequality by a positive value, the direction of the inequality remains the same. However, when multiplying or dividing both sides of the inequality by a negative value, the direction of the inequality changes. For example, if we have the inequality $\frac{1}{3}b \leq 8$, we can multiply both sides of the inequality by 3 to get $b \leq 24$. However, if we multiply both sides of the inequality by -3, the direction of the inequality changes, and we get $b \geq -24$.

Solving the Inequality $\frac{1}{3}b - 4 \leq 4$

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To solve the inequality $\frac{1}{3}b - 4 \leq 4$, we need to isolate the variable (b) on one side of the inequality sign. We can do this by adding 4 to both sides of the inequality to get $\frac{1}{3}b \leq 8$. Then, we can multiply both sides of the inequality by 3 to get $b \leq 24$.

Conclusion

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Solving linear inequalities is an essential skill in mathematics that can be used to solve various problems in algebra, geometry, and other branches of mathematics. By following the steps outlined in this article, you can solve linear inequalities with ease. Remember to be careful when multiplying or dividing both sides of the inequality by a negative value, as this can change the direction of the inequality.

Frequently Asked Questions

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Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form of an equation with a variable and a constant on one side of the inequality sign.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Q: What happens when I multiply or divide both sides of the inequality by a negative value?

A: When you multiply or divide both sides of the inequality by a negative value, the direction of the inequality changes.

Q: Can I use the same steps to solve non-linear inequalities?

A: No, non-linear inequalities require different steps to solve. Non-linear inequalities involve variables that are raised to a power or have a non-linear relationship with other variables.

Example Problems

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Problem 1

Solve the inequality $2x + 5 \leq 11$.

Solution

To solve the inequality $2x + 5 \leq 11$, we need to isolate the variable (x) on one side of the inequality sign. We can do this by subtracting 5 from both sides of the inequality to get $2x \leq 6$. Then, we can divide both sides of the inequality by 2 to get $x \leq 3$.

Problem 2

Solve the inequality $x - 3 \geq 7$.

Solution

To solve the inequality $x - 3 \geq 7$, we need to isolate the variable (x) on one side of the inequality sign. We can do this by adding 3 to both sides of the inequality to get $x \geq 10$.

Practice Problems

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Problem 1

Solve the inequality $\frac{1}{2}x - 2 \leq 3$.

Problem 2

Solve the inequality $x + 2 \geq 9$.

Problem 3

Solve the inequality $2x - 5 \leq 1$.

Problem 4

Solve the inequality $x - 1 \geq 6$.

Problem 5

Solve the inequality $\frac{1}{3}x + 2 \leq 5$.

References

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• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline
• [3] "Linear Algebra and Its Applications" by Gilbert Strang

Note: The references provided are for general information and are not specific to the topic of solving linear inequalities.

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Introduction

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In our previous article, we discussed the basics of solving linear inequalities, including what a linear inequality is, how to solve them, and some examples of solving linear inequalities. However, we know that there are many more questions that you may have about solving linear inequalities. In this article, we will answer some of the most frequently asked questions about solving linear inequalities.

Q&A

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Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form of an equation with a variable and a constant on one side of the inequality sign.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Q: What happens when I multiply or divide both sides of the inequality by a negative value?

A: When you multiply or divide both sides of the inequality by a negative value, the direction of the inequality changes.

Q: Can I use the same steps to solve non-linear inequalities?

A: No, non-linear inequalities require different steps to solve. Non-linear inequalities involve variables that are raised to a power or have a non-linear relationship with other variables.

Q: How do I know which direction to change the inequality when multiplying or dividing by a negative value?

A: When multiplying or dividing both sides of the inequality by a negative value, you need to change the direction of the inequality. For example, if you have the inequality $x \leq 5$ and you multiply both sides by -1, the inequality becomes $x \geq -5$.

Q: Can I use the same steps to solve inequalities with fractions?

A: Yes, you can use the same steps to solve inequalities with fractions. However, you need to be careful when multiplying or dividing both sides of the inequality by a fraction.

Q: How do I solve inequalities with fractions?

A: To solve inequalities with fractions, you need to follow the same steps as solving inequalities with whole numbers. However, you need to be careful when multiplying or dividing both sides of the inequality by a fraction.

Q: Can I use the same steps to solve inequalities with decimals?

A: Yes, you can use the same steps to solve inequalities with decimals. However, you need to be careful when multiplying or dividing both sides of the inequality by a decimal.

Q: How do I solve inequalities with decimals?

A: To solve inequalities with decimals, you need to follow the same steps as solving inequalities with whole numbers. However, you need to be careful when multiplying or dividing both sides of the inequality by a decimal.

Q: Can I use the same steps to solve inequalities with negative numbers?

A: Yes, you can use the same steps to solve inequalities with negative numbers. However, you need to be careful when multiplying or dividing both sides of the inequality by a negative number.

Q: How do I solve inequalities with negative numbers?

A: To solve inequalities with negative numbers, you need to follow the same steps as solving inequalities with whole numbers. However, you need to be careful when multiplying or dividing both sides of the inequality by a negative number.

Q: Can I use the same steps to solve inequalities with variables on both sides?

A: No, you cannot use the same steps to solve inequalities with variables on both sides. Inequalities with variables on both sides require different steps to solve.

Q: How do I solve inequalities with variables on both sides?

A: To solve inequalities with variables on both sides, you need to follow the steps of adding, subtracting, multiplying, or dividing both sides of the inequality by the same value. However, you need to be careful when multiplying or dividing both sides of the inequality by a negative value.

Example Problems

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Problem 1

Solve the inequality $2x + 5 \leq 11$.

Solution

To solve the inequality $2x + 5 \leq 11$, we need to isolate the variable (x) on one side of the inequality sign. We can do this by subtracting 5 from both sides of the inequality to get $2x \leq 6$. Then, we can divide both sides of the inequality by 2 to get $x \leq 3$.

Problem 2

Solve the inequality $x - 3 \geq 7$.

Solution

To solve the inequality $x - 3 \geq 7$, we need to isolate the variable (x) on one side of the inequality sign. We can do this by adding 3 to both sides of the inequality to get $x \geq 10$.

Problem 3

Solve the inequality $2x - 5 \leq 1$.

Solution

To solve the inequality $2x - 5 \leq 1$, we need to isolate the variable (x) on one side of the inequality sign. We can do this by adding 5 to both sides of the inequality to get $2x \leq 6$. Then, we can divide both sides of the inequality by 2 to get $x \leq 3$.

Practice Problems

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Problem 1

Solve the inequality $\frac{1}{2}x - 2 \leq 3$.

Problem 2

Solve the inequality $x + 2 \geq 9$.

Problem 3

Solve the inequality $2x - 5 \leq 1$.

Problem 4

Solve the inequality $x - 1 \geq 6$.

Problem 5

Solve the inequality $\frac{1}{3}x + 2 \leq 5$.

References

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• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline
• [3] "Linear Algebra and Its Applications" by Gilbert Strang

Note: The references provided are for general information and are not specific to the topic of solving linear inequalities.